Problem: Solve for $X$. $\left[\begin{array}{rr}3 & 4 & 7 \\ 2 & 5 & 1 \\8 &2 &5\end{array}\right]-X=\left[\begin{array}{rr}2 & 6 & 8 \\ 3 & 9 & 3 \\2 &5 &1\end{array}\right] $ $X=$
The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $\left[\begin{array}{rr}3 & 4 & 7 \\ 2 & 5 & 1 \\8 &2 &5\end{array}\right]-X=\left[\begin{array}{rr}2 & 6 & 8 \\ 3 & 9 & 3 \\2 &5 &1\end{array}\right] $ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}3 & 4 & 7 \\ 2 & 5 & 1 \\8 &2 &5\end{array}\right] ~~~~~~~~~ B = \left[\begin{array}{rr}2 & 6 & 8 \\ 3 & 9 & 3 \\2 &5 &1\end{array}\right]$ Then we can rewrite the equation as follows. $A-X=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}A-X&=B\\\\ A&=B+X\\\\ X&=A-B \end{aligned}$ Finding $X$ We found that $X=A-B$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=A-B \\\\&=\left[\begin{array}{rr}3 & 4 & 7 \\ 2 & 5 & 1 \\8 &2 &5\end{array}\right]-\left[\begin{array}{rr}2 & 6 & 8 \\ 3 & 9 & 3 \\2 &5 &1\end{array}\right] \\\\\\&=\left[\begin{array}{rr}(3-2) & (4-6) & (7-8) \\ (2-3) & (5-9) & (1-3) \\(8-2) &(2-5) &(5-1)\end{array}\right] \\\\\\&=\left[\begin{array}{rr}1 & -2 & -1 \\ -1 & -4 & -2 \\6 &-3 &4\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}1 & -2 & -1 \\ -1 & -4 & -2 \\6 &-3 &4\end{array}\right]$